Question

A line m is perpendicular to an angle bisector of ∠A. The sides of ∠A intersect this line m at points M and N. Prove that △AMN is isosceles.
PROVE with statement and reason
pls help thanks

Answer 1

Answer:

Step-by-step explanation:

Given:       A line m is perpendicular to the angle bisector of ∠A. We call this  

                 intersecting point as D. Hence, in figure ∠ADM=∠ADN =90°.

                 AD is angle bisector of ∠A. Hence, ∠MAD=∠NAD.

To Prove:   ΔAMN is an isosceles triangle. i.e any two sides in ΔAMN are

                   equal.

Solution:  Now, In ΔADM and ΔADN

                 ∠MAD=∠NAD     ...(1) (∵Given)

                  AD=AD                ...(2) (∵common side)

                  ∠ADM=∠ADN     ...(3) (∵Given)

                  Hence, from equation (1),(2),(3) ΔADM ≅ ΔADN

                                                         ( ∵ ASA  congruence rule)

                  ⇒ AM=AN

                  Now, In Δ AMN

                 AM=AN (∵ Proved)

                  Hence, ΔAMN is an isosceles  triangle.